Tucson, AZ to Bremerton, WA area: How quickly?

Have done the Wa. coast to Yuma in four days. That was pretty reasonable for late October. A lot will depend on the time of the year you are making your trips. I would not do it in any less then four days. Five would be more likely. That would be four nights worth of stop. Winter driving on 5 in Ore and Wa can be a bear sometimes.